Assume that $S$ is an inwardly oriented, piecewise-smooth surface with a piecewise-smooth, simple, closed boundary curve $C$ oriented negatively with respect to the orientation of $S$. Every point in $S$ has a nonzero, positive $z$. $ \oint_C (y\ln(z) \hat{\imath} + (z - 2x) \hat{k}) \cdot dr$ Use Stokes' theorem to rewrite the line integral as a surface integral. $ \iint_S ( $ $ \hat{\imath} + $ $\hat{\jmath} + $ $ \hat{k} ) \cdot dS$
Explanation: Assume we have a continuously differentiable three-dimensional vector field $F(x, y, z)$, an oriented piecewise-smooth surface $S$, and a piecewise-smooth, simple, closed boundary curve $C$ oriented positively with respect to $S$. Then Stokes' theorem states that we have the equality below: $ \oint_C F \cdot dr = \iint_S \text{curl}(F) \cdot dS$ If $C$ is negatively oriented, the line integral is equal to the negative of the double integral. [What does any of that mean?] When we use Stokes' theorem to translate from line integrals to surface integrals, we know $F$ and we want to find $\text{curl}(F)$. $\begin{aligned} F(x, y, z) &= y\ln(z) \hat{\imath} + (z - 2x) \hat{k} \\ \\ \text{curl}(F) &= \det \begin{pmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \\ \\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ \\ y\ln(z) & 0 & z - 2x \end{pmatrix} \\ \\ &= \left( \dfrac{y}{z} + 2 \right) \hat{\jmath} \\ \\ &+ (-\ln(z)) \hat{k} \end{aligned}$ Now that we know the curl of $F$, we can use it to find a surface integral equivalent to the original line integral. Because $C$ is negatively oriented, we put $-\text{curl}(F)$ inside the surface integral: $ \iint_S \left[ 0 \hat{\imath} + \left( -\dfrac{y}{z} - 2 \right) \hat{\jmath} + \ln(z) \hat{k} \right] \cdot dS$